electromagnetism - Biot-Savart law from Ampère's with multivariate calculus

Let us assume the validity of Ampère's circuital law

$$\oint_{\gamma}\mathbf{B}\cdot d\mathbf{x}=\mu_0 I_{\text{linked}}$$ where $\mathbf{B}$ is the magnetic field, $\gamma$ a closed path linking the current of intensity $I_{\text{linked}}$.

Can the Biot-Savart law

$$\mathbf{B}=\frac{\mu_0}{4\pi}\oint\frac{Id\boldsymbol{\ell}\times\hat{\mathbf{r}}}{r^2}=\frac{\mu_0}{4\pi}\int_a^b I\boldsymbol{\ell}'(t)\times\frac{\mathbf{x}-\boldsymbol{\ell}(t)}{\|\mathbf{x}-\boldsymbol{\ell}(t)\|^3}dt$$ where $\boldsymbol{\ell}:[a,b]\to\mathbb{R}^3$ is a parametrisation of a closed (or infinite) wire carrying the current $I$, be inferred without using Dirac's $\delta$, by using the tools of multivariate calculus and elementary differential geometry only, at least if we assume the validity of the Gauss law for magnetism or other of the Maxwell equations? All the proofs I have found (such as this, where, as far as I understand, $$\nabla^2\left[\frac{\mu_0}{4\pi}\int\frac{\mathbf{J}(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}d^3r'\right]=-\mu_0\mathbf{J}(\mathbf{r})$$ is derived by using the $\delta$) use the expression $$\mathbf{B}=\frac{\mu_0}{4\pi}\int_V\frac{\mathbf{J}\times\hat{\mathbf{r}}}{r^2}d^3x$$ and Dirac's $\delta$, but I wonder whether, both assuming a linear current distribution as when we use the expression of the magnetic field as $$\mathbf{B}=\frac{\mu_0}{4\pi}\oint\frac{Id\boldsymbol{\ell}\times\hat{\mathbf{r}}}{r^2}$$ and assuming a tridimensional spatial current distribution, it is possible to prove the Biot-Savart law from Ampère's without the use of the $\delta$. I heartily thank any answerer.